### Experimental outcomes

Determine 1d reveals an optical micrograph of an AB-stacked tetralayer graphene pattern that consisted of encapsulated graphene with a prime and a backside gate electrode. The graphene pattern is encapsulated with two *h-*BN flakes with completely different crystallographic orientations. Relative angle of crystal axis between graphene and backside *h-*BN flake was set to be roughly parallel (really 0.35 levels) which varieties moiré superlattice. Alternatively, the angle between prime *h*-BN and graphene was about 16 levels and the moiré construction resulting from prime *h-*BN is negligible. The low-temperature electrical mobility of this system was above 4 × 10^{4} cm^{2}V^{−1}s^{−1}. As proven in Fig. 1e, peaks which come up from moiré superlattice seem within the plot of resistivity (*ρ*) in opposition to backside gate voltage (*V*_{b}). Along with the clear principal peak at *V*_{b} ≈ 0 V, a few aspect peaks are discernible at *V*_{b} ≈ 40 and −45 V, that are paying homage to superlattice peaks as a result of secondary Dirac factors in a moiré superlattice fashioned with *h*-BN and monolayer (or bilayer) graphene.

The resistivity traces (*ρ* vs *V*_{b}) confirmed important variation with prime gate voltage (Fig. 2a), and detailed measurements of the resistivity map revealed a fantastic construction of resistivity peaks (Fig. 2b). The straight ridge from the highest left to backside proper of Fig. 2b is the situation of cost neutrality, which is given by *n*_{tot} = 0, the place *n*_{tot} is the service density induced by the highest and backside gate voltage,

$$n_{{mathrm{tot}}} = (C_{mathrm{t}}V_{mathrm{t}} + C_{mathrm{b}}V_{mathrm{b}})/e.$$

(1)

Right here, *C*_{t} and *C*_{b} are the particular capacitances for the highest and backside gate electrodes. The perpendicular electrical flux density (*D*_{⊥}), which is roughly proportional to the power of the perpendicular electrical area, could be calculated as

$$D_ bot = (C_{mathrm{t}}V_{mathrm{t}} – C_{mathrm{b}}V_{mathrm{b}})/2.$$

(2)

Determine 2c is a replot of the map. One can see ridges of resistivity (bb+, bb−, MDP) that have been not too long ago present in AB-stacked tetralayer graphene^{31,32,33,43,44}. Ridges bb+ and bb- mirror the bottoms of the light-mass bilayer-like bands, whose dispersions are almost flat in a perpendicular electrical area. MDP is because of mini-Dirac cones. Nonetheless, the opposite ridges (A+, B+, C+, and so forth.) are absent from pristine AB-stacked tetralayer graphene.

The peaks within the hint of the gate voltage dependence of resistivity are paying homage to superlattice peaks within the moiré superlattice of *h*-BN/monolayer graphene: one can calculate the interval of the moiré potential^{3,4,5,6,7} from the dimensions of the superlattice Brillouin zone, which could be estimated from the service density of the resistivity peak as a result of superlattice (*n*_{FBZ})^{3,4,5,6,7}. The mismatch angle *ϕ* could be estimated as^{3,4,5,6,7},

$$n_{FBZ} = frac{8}{{sqrt 3 lambda ^2}}$$

(3)

the place

$$lambda = frac{{left( {1 + delta } proper)a}}{{sqrt {2left( {1 + delta } proper)left( {1 – {mathrm{cos}}phi } proper) + delta ^2} }}.$$

(4)

Right here, *λ* is the wave size (or lattice fixed) of the moiré potential, *δ* is the distinction between the lattice constants of *h*-BN and graphene, and *a* is the lattice fixed of graphene^{3}. As a result of the tetralayer graphene has two low power bands, *n*_{FBZ} can’t typically be decided from service density of the superlattice within the hint of gate voltage *vs*. resistivity. Nonetheless one can roughly estimate *λ* if one ignore contribution of the smaller Fermi floor. In our case, the decrease restrict of λ was about 12.4 nm utilizing (|n_{mathrm{tot}}|) of A+ (A−) ridge at (|D_ bot | = 0). Corresponding mismatch angle was estimated to be (phi approx 0.52^ circ). Alternatively the higher restrict is the right aligned case which is about 14 nm, in order that *λ* of our tetralayer graphene moire superlattice lies between them. The estimated values are near the experiment in monolayer graphene^{3,7,9,10}.

The Landau ranges have been additionally influenced by the moiré construction. Determine 2d reveals a map of longitudinal resistivity (*ρ*_{xx}), which was measured as a perform of *n*_{tot} and magnetic area (*B*) for (D_ bot = 0,{mathrm{cm}}^{ – 2}{mathrm{As}}). Every streak indicated by vivid and darkish strains is a Landau degree and the power hole between them. The Landau degree construction between (n_{{mathrm{tot}}} = – 2 occasions 10^{12},{mathrm{and}},2 occasions 10^{12}) cm^{−2} are particular to pristine AB-stacked tetralayer graphene^{31,32,33,45} which ensured that our pattern was AB-stacked tetralayer. As well as, at (n_{{mathrm{tot}}} sim 3.5 occasions 10^{12},{mathrm{cm}}^{ – 2}), one can see a Landau fan paying homage to that for the secondary Dirac cones within the case of moiré superlattice in bilayer graphene. Round this service density, ridge A+ seems (Fig. 2c).

### Numerical simulation

Though Ridges A+(A−), B+(B−), and C+(C−) could be the satellite tv for pc peaks just like these within the moiré superlattice for mono- and bilayer graphene, they confirmed uncommon response to the perpendicular electrical area: the ridges exhibited important variation with respect to (|D_ bot |). Such conduct was not noticed within the case of bilayer graphene (see Supplementary Word). To review the origin, we in contrast the outcomes of the experiment with a theoretical calculation. Determine 2e reveals a map of resistivity that was numerically calculated from the dispersion relations by utilizing Boltzmann transport idea with the fixed rest time approximation (see Supplementary Methodology). The band construction calculation took account of an efficient moiré potential based mostly on the *h-*BN-graphene hopping mannequin introduced by Moon and Koshino^{24}. The amplitude of the efficient potential and mismatch angle of the crystal axes between *h*-BN and graphene have been adjusted in order that the calculation reproduced the experimental outcomes (see Supplementary Dialogue). The experimental map was roughly reproduced for *θ* = 0.35°. The interval of the moiré potential was 13.0 nm (about 53 occasions the lattice fixed of graphene). This worth is barely completely different from the tough estimation (*λ* = 12.4 nm, *θ* = 0.52°) The calculation roughly reproduced the experimental map of resistivity, which signifies that the moiré potential considerably influenced the band construction of the multilayer graphene regardless that the potential is current on the first layer of graphene contacting *h*-BN. The ridges bb+ and bb− are intrinsic to AB-stacked tetralayer graphene^{31,32,43,44}. By finding out the relation between the form of the Fermi floor (the power contour of the dispersion relation) and service density, it was discovered that the opposite ridges appeared when an power hole related to the superlattice potential opened. These resistivity peaks (ridges) are resulting from formation of an power hole related to overlapping of the Fermi surfaces that are translated by reciprocal vectors within the prolonged zone scheme, as proven in Fig. 3a–c. These present the numerically calculated Fermi floor(s) for various energies within the case of (left| {D_ bot } proper| = 3.2 occasions 10^{12},{mathrm{cm}}^{ – 2}{mathrm{As}}). Reciprocal vectors of the moiré potential is proven in Fig. 3d. The Fermi floor of AB-stacked tetralayer graphene consists of trigonally warped circles. It modifications its topology by opening an power hole when the Fermi surfaces are overlapped within the superlattice potential as proven schematically in Fig. 3e–g. Opening an power hole reduces the digital states accessible for electrical conduction, and thereby reduces conductivity. Ridge A+ (A−) are for overlapping of the Be (Bh) band. (Be (Bh) is an electron-like (hole-like) heavy mass bilayer-like band; be (bh) is an electron-like (hole-like) mild mass bilayer-like band.) Ridges B+ (B−) and C+ (C−) are for overlapping of Be (Bh) and be (bh), which happen at completely different service densities (energies) due to the trigonally warped Fermi floor (Fig. 3b–c, f, g). For larger service densities, overlapping be (bh) with itself may happen and it ought to end in a corresponding resistance ridge.

Determine 3h–j present simplified band diagrams within the prolonged zone scheme to clarify the opening of the power hole; FBZ is the primary Brillouin zone, and similar band diagrams are translated by reciprocal vectors. Small power gaps open when bands cross one another, and this ends in resistance ridges A+, B+, C+, and so forth.; for instance, ridge A+ happens when band Be crosses itself within the neighboring zone; ridge B+ happens when be crosses Be, *and so forth*. As a result of the band construction is trigonally warped^{28,29}, the power gaps related to the overlapping typically happen at wave numbers apart from on the Brillouin zone boundary, as could be seen within the determine.

The difficult dependence of the peaks (ridges) on *n*_{tot} and (D_ bot) is as a result of multiband property related to the variation of dispersion relations by means of the perpendicular electrical area. The power gaps amongst be, Be, Bh and bh widen with rising (|D_ bot |), as proven in Fig. 3h–j. Accordingly a chemical potential within the neighborhood of the power hole for peak A+ (A−) (the dashed line), typically varies by (|D_ bot)|. Nonetheless, the form of the fermi surfaces of Be (Bh) doesn’t fluctuate largely at that power, in order that service density in Be (Bh) band (for the power) is roughly unchanged by (|D_ bot)|. Alternatively, the distinction in power between the underside of be and the underside of Be will increase with rising (|D_ bot)|, and Ridge A+ (A−) bends to smaller |*n*_{tot}| with rising (|D_ bot)| as a result of the variety of carriers in be decreases (Fig. 3h, i). Above the essential (|D_ bot)|, the power of the underside of be (bh) turns into bigger than the chemical potential (dashed line), and solely the Fermi floor of Be (Bh) is current (Fig. 3j). This transition could be seen within the map (Fig. 2c, e) as crossings of ridge A+ (A−) and bb+ (bb−) at (left| {n_{{mathrm{tot}}}} proper| sim 2.3 occasions 10^{12}{mathrm{cm}}^{ – 2})and (left| {D_ bot } proper| sim 3.5 occasions 10^{ – 7},{mathrm{cm}}^{ – 2}{mathrm{As}}). The electron-hole asymmetry within the map could be equally defined by contemplating the electron-hole asymmetry within the dispersion relation. The Fermi floor areas of the light-mass bilayer-like band and the heavy-mass one have completely different ratios in electron and gap regimes due to this asymmetry.

As proven above, the resistance peaks (ridges) as a result of moiré superlattice could be defined by power gaps opening. It’s generally accepted that power gaps type on the Brillouin zone boundary of the superlattice. Certainly, the power hole is predicted to open on the Brillouin zone boundary in moiré superlattice for monolayer graphene which is roughly isotropic. Nonetheless, it could open contained in the Brillouin zone in an anisotropic electron system (Fermi floor) as within the case of AB-stacked graphene with a number of layers. Textbooks on solid-state physics inform us that, within the presence of a periodic potential, an power hole opens through interference of wave capabilities (phi left( {mathbf{ok}} proper)) and (phi ({mathbf{ok}} pm {mathbf{G}})), the place **ok** is a wave vector and **G** is likely one of the reciprocal vectors. If the unique band construction satisfies (Eleft( {mathbf{ok}} proper) = Eleft( {{mathbf{ok}} pm {mathbf{G}}} proper)), an power hole ought to open on a boundary of the Brillouin zone. Alternatively, if the unique band is anisotropic, as within the case of AB-stacked multilayer graphene, the situation, (E({mathbf{ok}}) = E({mathbf{ok}} pm {mathbf{G}})), could be glad and an power hole ought to type regardless that **ok** shouldn’t be on the boundary of however quite contained in the Brillouin zone, as could be seen in Fig. 3a. Comparable phenomena arising from anisotropy have not too long ago been mentioned in photonic crystals^{46} and phonic crystals^{47,48,49}; Bragg reflection happens contained in the Brillouin zone in anisotropic media, quite than on the boundary. As well as, an power hole opens when the Fermi surfaces of various bands are overlapped. The wave capabilities in band 1, (phi _1({mathbf{ok}}_1)), and in band 2, (phi _2({mathbf{ok}}_2)), that are orthogonal to one another within the absence of moiré potential, anti-cross when (E({mathbf{ok}}_1) = E({mathbf{ok}}_2 pm {mathbf{G}})) is glad. This may be seen in Fig. 3b, c. Power gaps open and the Fermi surfaces change their topology when be and Be are overlapped.

### Efficient moiré potential

The detailed construction of the numerically calculated dispersion relation is very depending on the fashions of the efficient moiré potential and its amplitude. Nonetheless, the important characteristic of the Fermi floor topology is unchanged by the selection of mannequin. We calculated the dispersion relations and resistivity maps for *h*-BN-graphene hopping fashions^{22,23,24,50}, the 2D cost modulation mannequin^{51}, and the potential modulation mannequin^{3,22,50} (see Supplementary Dialogue). The *h*-BN-graphene hopping fashions roughly reproduce the experimental resistivity map if the potential amplitude is lower than about half of these given within the theoretical works^{22,23,24,50}. Utilizing any of those fashions, the resistivity map was roughly reproduced for a small enough potential amplitude, which signifies that the topological transitions of the Fermi floor resulting from overlapping happen whatever the mannequin used.

### Trigonal warping

If the resistivity peaks (ridges) originate from overlapping of the Fermi floor, the resistivity maps ought to present a big dependence of the trigonal warping as a result of the overlapping is considerably influenced by the form of the Fermi floor. Determine 4a–c reveals the maps calculated for various values of *γ*_{3} of the SWMcC parameters, which tunes the trigonal warping; *γ*_{3} = 0 eV is for round Fermi surfaces; *γ*_{3} = 0.3 eV approximates the experiment. Overlapping of Be and be (Bh and bh) can happen at completely different factors, and thereby, at completely different energies (or service densities) within the trigonally warped case as illustrated in Fig. 3f–h. This ends in separate ridges B+ (B−) and C+ (C−) (*γ*_{3} = 0.3 eV; Fig. 4c). With lowering *γ*_{3}, ridges B+ and C+ (B− and C−) merge right into a single ridge (*γ*_{3} = 0 eV; Fig. 4a). On this case, the power hole ought to open on the boundary of the superlattice Brillouin zone.